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\begin{document}

Let $\eta$ be the prior information,$e_{1:t}$ be the observations made up to time $t$, $e_t \in \{-1,+1\}$ , $c \in [0,1]$ be the coherence, and $d \in \{+1, -1\}$ be the hidden direction. We have the following graphical model:
\begin{center}
  $
\psmatrix[mnode=circle, colsep=2cm, rowsep=2cm]
 & \eta \\
 c & d\\
 e_{1:t} & e_{t+1}
\ncline{->}{1,2}{2,2}
\ncline{->}{2,1}{3,1}
\ncline{->}{2,2}{3,1}
\ncline[linestyle=dashed]{->}{2,1}{3,2}
\ncline[linestyle=dashed]{->}{2,2}{3,2}
\endpsmatrix
$
\end{center}

Upon observing $e_{t+1}$, the belief over $d$ will be updated as:
\begin{eqnarray*}
  \Pr{d|e_{1+t+1}, \eta} &=& \int_c \Pr{d,c|e_{1:t+1}, \eta} dc \\
    &=& \frac{\int_c \Pr{e_{t+1}|c,d} \Pr{c,d|e_{1:t},\eta} dc }{\Pr{e_{t+1} | e_{1:t},\eta}}\\
&=&   \frac{\int_c \Pr{e_{t+1}|c,d} \Pr{c|d, e_{1:t}} dc \Pr{d|e_{1:t},\eta}}{\Pr{e_{t+1} | e_{1:t},\eta}}\\
&=&  \frac{ \Pr{e_{t+1}|d, e_{1:t}}\ \ \Pr{d|e_{1:t},\eta}}{\sum_d\Pr{e_{t+1} |  d } \Pr{d | e_{1:t},\eta}} \\
b_{t+1}(d) &=& \frac{ \Pr{e_{t+1}|d, e_{1:t}}}{\sum_d \Pr{e_{t+1} | d} b_t(d)} b_t(d)
\end{eqnarray*}

Therefore, the transition probability $\Pr[b_{t+1} | b_{t}]$ is independent of the prior $\eta$. And more importantly, it becomes non-Markov. Belief over $c$ alone are updated in a similar way.

{\bf Limitations of Beta distribution}
\begin{itemize}
\item  All beta distributions are uni-modal.  
\item Fortunately, a mixture of beta distributions is still the conjugate prior of binomial distribution. But the weights over each mode will be changed. Let the prior be $w_1Beta(\alpha_1,\beta_1) + w_2 Beta(\alpha_2, \beta_2)$, and observations be $n$ heads and  $m$ tails, then the posterior becomes $w_1' Beta(\alpha_1 + n, \beta_1 + m) + w_2' Beta(\alpha_2 + n, \beta+ m)$
\end{itemize}


\end{document}